A179994 a(n) = 11*a(n-2) - 8*a(n-4), starting a(0)=0, a(1)=1, a(2)=5, a(3)=12.

0, 1, 5, 12, 55, 124, 565, 1268, 5775, 12956, 59005, 132372, 602855, 1352444, 6159365, 13817908, 62930175, 141177436, 642957005, 1442408532, 6569085655, 14737074364, 67116286165, 150568549748, 685726462575, 1538357452316

Offset

0,3

Comments

The limiting ratio a(n)/a(n-1) alternates between 2.2433981132057377... and 4.554247641507091...

Links

Table of n, a(n) for n=0..25.

Eric Weisstein's World of Mathematics, Symplectic Group.

Index entries for linear recurrences with constant coefficients, signature (0,11,0,-8).

Formula

Given the four matrices m(1) = {{1, 0}, {0, 1}}, m(2) = {{1, 1}, {0, 1}}, m(3) = {{1, 0}, {1, 1}}, and m(4) = m(1)+m(2)+m(3), repeatedly multiply the vector {0,1} with m(1), m(2) etc in a round-robin periodic fashion. a(n) is the top value of the vector after each second multiplication.

G.f.: x*(1 + 5*x + x^2) / (1 - 11*x^2 + 8*x^4). - R. J. Mathar, Feb 10 2011

a(n) = 5*a(n-1) - a(n-2) if n is even. a(n) = 4*a(n-1) - 8*a(n-2) if n is odd. - R. J. Mathar, Jun 18 2014

Example

Multiplication with m1 updates the vector to {1,1}, then with m2 to {5,7}, then with m3 to {5,7}, then with m4 to {12,7}, then with m1 {12,19}, then with m2 {55,69}, ... The leading components are 1, 1, 5, 5, 12, 12, 55, 55, ... and each second defines the sequence.

Maple

with(LinearAlgebra) ; Am := [ [1, 0, 0, 1], [1, 1, 0, 1], [1, 0, 1, 1], [3, 1, 1, 3] ]: v := Vector[column](2, [0, 1]) :
for l from 0 to 10 do
        A := Matrix(2, 2, op(1+(l mod 4), Am) ) ; v := A.v;
        if type(l, 'even') then print(v[1]) ;  end if;
end do: # R. J. Mathar, Feb 10 2011

Mathematica

a[1] = {{1, 0}, {0, 1}};
a[2] = {{1, 1}, {0, 1}};
a[3] = {{1, 0}, {1, 1}};
a[4] = a[1] + a[2] + a[3]
v[0] = {0, 1};
v[n_] := v[n] = a[1 + Mod[n, 4]].v[n - 1]
Table[v[n][[1]], {n, 0, 60, 2}]

Crossrefs

Sequence in context: A062978 A189421 A066961 * A131549 A342549 A346746

Adjacent sequences: A179991 A179992 A179993 * A179995 A179996 A179997

Keyword

nonn,easy

Author

Roger L. Bagula, Aug 05 2010

Status

approved